Multiplication is the coordination of (at least) two levels of units, a grouping unit and a number of units to be grouped.
In other words, one basic concept of multiplication is the idea of repeated equal groups. However, that is not the only concept! What are some others? [posting opp]
Expect the repeated addition, there are some other concept of multiplication, such as the number of elements in the union of “a” disjoint equivalent sets, each containing “b” elements.
For example, multiplication by joining segments of equal length on a number line.
4 x 3 = 12, we can say “4” is the number of segments being joined, and “3” is the length of one segment.
We also can use the area of rectangle to explain multiplication,
Two units in multiplication are the width and length in area of rectangle. ( YuanNiu)
Strategies for Equal Groups Problems
We began developing strategies for whole number multiplication in base twelve the way that children do--by using counting and additive strategies. The strategies we developed--and saw children use--are:
[posting opportunity: Describe each strategy below with an example in base twelve.]
Coordinating Two Counts:
Strategic Counting:
Doubling: The reason doubling is so useful is because working with smaller numbers is easier to multiply with than larger numbers. Lets take the multiplication problem 7 times 8. For example if you double (7) in base 12...you get (12) in base twelve. Then you just multiplied 7 and 2. If you double that you get (24) in base 12. If you double that again you get (48) in base 12. Since you just used 8 7s you solved the problem 7 times 8. However, it was made much simpler because of the way you doubled it 3 times using 8 sevens. ~Lindsey Heller
Conversion to Bases and Ones:
Distributive Reasoning (Rounding is included here):[give two examples here, one that involves rounding and one that does not; explain why rounding is included in distributive reasoning]
We also discussed Commutative Reasoning in Multiplication. Explain the main idea of commutative reasoning in multiplication. [posting opp! See #4 of the blue sheet...]
Then we examined whether we could use our strategies with division problems...
What is whole number division?
To do this, we had to think about what division is. There are two kinds of division based on equal groups situations. Can you explain them both? [posting opp!]
There are two types of division problems that are based on equal groups’ multiplication problems: measurement and sharing.
(# of items in a group) x (# of groups) = (total # of items)
In an equal group’s multiplication problem, the total number of items is unknown. In measurement division problems, the number of groups is unknown. In sharing division, the group size is unknown. For example, “you have a total of 52 cards, with 13 cards in each stack. How many stacks of 13 cards are there?” In this problem, the number of group is unknown, it is the measurement division problem. “You have total of 52 cards and there are 4 stacks, how many cards in each stack?” the number of items in a group is unknown in this problem, so it is the sharing division problem. ( YuanNiu)
Strategies for Measurement Division Problems
Our strategies for multiplication worked in measurement division problems, although they did not always work exactly in the same way. The strategies we developed--and saw children use--are: [posting opportunity: Describe each strategy below with an example in base twelve.]
Coordinating Two Counts:
Strategic Counting:
Doubling:
Conversion to Groups Other than Bases:
Inverse of the Distributive Property:
Rounding:
Rounding with division is really useful for getting approximate answer as opposed to an exact answer. Let’s take the measurement division problem (T1)12 divided by (E)12 . in base twelve, (E)12 is close to the base number, so we rounding it to dozen which is add 1 more, it’s easy to get the answer because we are dividing by the base number (10)12. Depending on problems (if you rounding the size of groups bigger, then the number of groups become smaller) you need to think whether your answer is too high or too low. We have T # of (10)12 and (1)12 made of (T1)12 . Because we add T # of 1 which is means we add (T)12 more , the size of groups are bigger which makes the number of groups smaller. So the totally number we get is T # of (E)12 , (T)12 and (1)12. We add (T)12 and (1)12 together is (E)12, then plus T# of (E)12 we have E # of ( E)12. ( YuanNiu)
Strategies for Sharing Division Problems
Sharing division problems are "thornier" to solve than measurement division problems. Why? [posting opp!]
We looked at two strategies that children often use to solve these problems, but neither is that efficient for our own use. So, then we developed a third strategy. [posting opportunity: Describe each strategy below with an example in base twelve. For the each of the first two strategies, tell what one elementary school student did to solve their problem.]
Dealing by Ones:
We saw Alex use this strategy.
Strategic Trials:
We saw both Vanessa and Victoria use this strategy.
Using Commutative Reasoning: [explain this strategy well, telling why it's important if we want to be able to use the strategies for measurement division in solving sharing division problems!]
In sharing problems, we know the total number of items and the number of groups, the group size is unknown. In order to solve the problem, we need to count by number of groups till the final answer is approach to total number of items, then the number of groups is the times that we count by the number of groups. This is commutative reasoning.
For example: A baker bakes (48)12 cookies on (7)12 pans. How many cookies are on each pan?
We are counting by 7s to solve the problem. This is because there are 7 pans to put cookies on, we know that there are an equal number of cookies on each sheet, and every time you add one to each pan, you are adding 7 to the total amount of cookies that you have, so you can count by 7’s in this problem until you get to (48)12. Counting by 7’s is adding one cookie to each seven pans.
Every time when we counting by number of groups, it’s means we add certain count numbers to groups. If we have 1 in each group, then we have 7 in 7 groups; if we have 2 in each group, then we have 7 + 7= 7 x 2 = (12)12in 7 groups, etc.
It’s important if we want to be able to use the strategies for measurement division in solving sharing division problems. Because every time when we are counting, we put certain amount of numbers in certain groups and adding up till we get to total number. (YuanNiu)
On Strategic Multiplicative Reasoning
What is whole number multiplication?
Multiplication is the coordination of (at least) two levels of units, a grouping unit and a number of units to be grouped.
In other words, one basic concept of multiplication is the idea of repeated equal groups. However, that is not the only concept! What are some others? [posting opp]
Expect the repeated addition, there are some other concept of multiplication, such as the number of elements in the union of “a” disjoint equivalent sets, each containing “b” elements.
For example, multiplication by joining segments of equal length on a number line.
4 x 3 = 12, we can say “4” is the number of segments being joined, and “3” is the length of one segment.
We also can use the area of rectangle to explain multiplication,
Two units in multiplication are the width and length in area of rectangle. ( YuanNiu)
Strategies for Equal Groups Problems
We began developing strategies for whole number multiplication in base twelve the way that children do--by using counting and additive strategies. The strategies we developed--and saw children use--are:
[posting opportunity: Describe each strategy below with an example in base twelve.]Coordinating Two Counts:
Strategic Counting:
Doubling:
The reason doubling is so useful is because working with smaller numbers is easier to multiply with than larger numbers. Lets take the multiplication problem 7 times 8. For example if you double (7) in base 12...you get (12) in base twelve. Then you just multiplied 7 and 2. If you double that you get (24) in base 12. If you double that again you get (48) in base 12. Since you just used 8 7s you solved the problem 7 times 8. However, it was made much simpler because of the way you doubled it 3 times using 8 sevens.
~Lindsey Heller
Conversion to Bases and Ones:
Distributive Reasoning (Rounding is included here): [give two examples here, one that involves rounding and one that does not; explain why rounding is included in distributive reasoning]
We also discussed Commutative Reasoning in Multiplication. Explain the main idea of commutative reasoning in multiplication. [posting opp! See #4 of the blue sheet...]
Then we examined whether we could use our strategies with division problems...
What is whole number division?
To do this, we had to think about what division is. There are two kinds of division based on equal groups situations. Can you explain them both? [posting opp!]
There are two types of division problems that are based on equal groups’ multiplication problems: measurement and sharing.
(# of items in a group) x (# of groups) = (total # of items)
In an equal group’s multiplication problem, the total number of items is unknown. In measurement division problems, the number of groups is unknown. In sharing division, the group size is unknown. For example, “you have a total of 52 cards, with 13 cards in each stack. How many stacks of 13 cards are there?” In this problem, the number of group is unknown, it is the measurement division problem. “You have total of 52 cards and there are 4 stacks, how many cards in each stack?” the number of items in a group is unknown in this problem, so it is the sharing division problem. ( YuanNiu)
Strategies for Measurement Division Problems
Our strategies for multiplication worked in measurement division problems, although they did not always work exactly in the same way. The strategies we developed--and saw children use--are:[posting opportunity: Describe each strategy below with an example in base twelve.]
Coordinating Two Counts:
Strategic Counting:
Doubling:
Conversion to Groups Other than Bases:
Inverse of the Distributive Property:
Rounding:
Rounding with division is really useful for getting approximate answer as opposed to an exact answer. Let’s take the measurement division problem (T1)12 divided by (E)12 . in base twelve, (E)12 is close to the base number, so we rounding it to dozen which is add 1 more, it’s easy to get the answer because we are dividing by the base number (10)12. Depending on problems (if you rounding the size of groups bigger, then the number of groups become smaller) you need to think whether your answer is too high or too low. We have T # of (10)12 and (1)12 made of (T1)12 . Because we add T # of 1 which is means we add (T)12 more , the size of groups are bigger which makes the number of groups smaller. So the totally number we get is T # of (E)12 , (T)12 and (1)12. We add (T)12 and (1)12 together is (E)12, then plus T# of (E)12 we have E # of ( E)12. ( YuanNiu)
Strategies for Sharing Division Problems
Sharing division problems are "thornier" to solve than measurement division problems. Why? [posting opp!]
We looked at two strategies that children often use to solve these problems, but neither is that efficient for our own use. So, then we developed a third strategy.[posting opportunity: Describe each strategy below with an example in base twelve. For the each of the first two strategies, tell what one elementary school student did to solve their problem.]
Dealing by Ones:
We saw Alex use this strategy.
Strategic Trials:
We saw both Vanessa and Victoria use this strategy.
Using Commutative Reasoning: [explain this strategy well, telling why it's important if we want to be able to use the strategies for measurement division in solving sharing division problems!]
In sharing problems, we know the total number of items and the number of groups, the group size is unknown. In order to solve the problem, we need to count by number of groups till the final answer is approach to total number of items, then the number of groups is the times that we count by the number of groups. This is commutative reasoning.
For example: A baker bakes (48)12 cookies on (7)12 pans. How many cookies are on each pan?
We are counting by 7s to solve the problem. This is because there are 7 pans to put cookies on, we know that there are an equal number of cookies on each sheet, and every time you add one to each pan, you are adding 7 to the total amount of cookies that you have, so you can count by 7’s in this problem until you get to (48)12. Counting by 7’s is adding one cookie to each seven pans.
Every time when we counting by number of groups, it’s means we add certain count numbers to groups. If we have 1 in each group, then we have 7 in 7 groups; if we have 2 in each group, then we have 7 + 7= 7 x 2 = (12)12in 7 groups, etc.
It’s important if we want to be able to use the strategies for measurement division in solving sharing division problems. Because every time when we are counting, we put certain amount of numbers in certain groups and adding up till we get to total number. (YuanNiu)